Today I think I repaired my front bike inner tube properly. I got a bike holder to put my bike on while working on it, and after I set it up, I thought I could take another look at the inner tube. Turns out the patch got lose again, probably during reassembly or while adding air.
I also found out that the tube that I got from the shop is only a 24/28-622 size tube, which is very small for a 45-622 wheel. So I probably inflated it too much to fill the wheel, making a defect more likely.
I will see if it holds the air over night but I’ll probably switch out the inner tube to the right size one once I get it tomorrow. This is fun to figure out, even if a bit frustrating. But better at home in the garage than on the track.